Source: https://jmrplens.github.io/phonometry/reference/api/materials/slow-sound-absorber/ Slow-sound slit panels loaded with Helmholtz resonators (perfect absorbers). A rigid panel perforated by a periodic array of thin closed slits, whose upper wall is loaded by an array of Helmholtz resonators (HRs), behaves as a deep-subwavelength, locally reacting sound absorber. The resonators slow the sound inside the slit, pulling the slit resonance down to the deep subwavelength regime, and the intrinsic visco-thermal losses can be tuned to exactly balance the leakage of the structure (critical coupling), giving perfect absorption at a chosen frequency and angle. The model follows the transfer-matrix treatment of Jimenez, Groby, Pagneux and Romero-Garcia (*Iridescent Perfect Absorption in Critically-Coupled Acoustic Metamaterials Using the Transfer Matrix Method*, Appl. Sci. 2017, 7, 618) together with the resonator model and end corrections detailed in the supplementary material of Jimenez, Huang, Romero-Garcia, Pagneux and Groby (*Ultra-thin metamaterial for perfect and quasi-omnidirectional sound absorption*, Appl. Phys. Lett. 2016, 109, 121902). The building blocks, all in the `e^{+j w t}` convention used throughout phonometry (a passive medium has `Im(k) < 0`): * **Visco-thermal effective parameters.** The slit of height `h` uses the narrow-channel effective density and bulk modulus (Appl. Sci. Eq. (6); Appl. Phys. Lett. Eqs. (A1)-(A2)): `rho_s = rho0 [1 - tanh((h/2) G_rho) / ((h/2) G_rho)]^-1` and `kappa_s = kappa0 [1 + (gamma - 1) tanh((h/2) G_kappa) / ((h/2) G_kappa)]^-1` with `G_rho = sqrt(j w rho0 / eta)` and `G_kappa = sqrt(j w Pr rho0 / eta)`. The square necks and cavities use the rectangular-duct series of Stinson (1991), reproduced as Appl. Sci. Eqs. (7)-(8) with the transverse wavenumbers `alpha_k = (2k+1) pi / a` and `beta_m = (2m+1) pi / b`. The duct series is printed in the opposite time convention of the source; it is returned conjugated here so the neck and cavity share the `e^{+j w t}` passivity of the slit. Both models are pinned in the tests to their exact limits: the effective density tends to `rho0` and the bulk modulus to `kappa0` as the boundary layers vanish, and `j w rho` tends to the Poiseuille flow resistivity of the channel as `w -> 0` (`12 eta / h^2` for the slit, `28.454 eta / w^2` for a square duct). * **Helmholtz-resonator impedance.** Each resonator is a neck (length `l_n`, side `w_n`) over a closed cavity (length `l_c`, side `w_c`); its impedance follows Appl. Phys. Lett. Eq. (A23) with the neck-to-cavity radiation end correction of Eqs. (A24)-(A26). * **Transfer matrix.** The panel is the chain `M_dl (M_s M_HR M_s)...` of half-lattice slit steps (Appl. Sci. Eq. (2)), resonators as point shunt scatterers (Eq. (3)) and the slit-radiation end correction (Eq. (3)/(A27)). The slit-radiation series impedance is printed in the sources as `-i w dl_slit rho0 / (phi_t S0)`; like the duct series it is used conjugated here (`+j w`), so it acts as the added radiation mass it models and lowers the slit-panel resonance. The rigidly-backed reflection factor is `R = (T11 cos(theta) - Z0 T21) / (T11 cos(theta) + Z0 T21)` with `Z0 = rho0 c0 / S0` (Eq. (4)), and `alpha = 1 - |R|^2`. Perfect absorption (critical coupling) is reached when the reflection zero sits on the real-frequency axis, i.e. `Re(Z) cos(theta) = Z0` and `Im(Z) = 0` with `Z = T11 / T21` the acoustic surface impedance (Eq. (9)). > Auto-generated from the source docstrings by `scripts/generate_api_docs.py` (`make api-docs`). Do not edit by hand. ## critical_coupling_design ```python critical_coupling_design( target_frequency: float, resonator: HelmholtzResonator, *, lattice_step: float, period: float, angle: float = 0.0, slit_height_bounds: tuple[float, float] = (0.0002, 0.005), cavity_length_bounds: tuple[float, float] = (0.002, 0.2), end_correction: bool = True, slit_radiation: bool = True, speed_of_sound: float = 343.0, air_density: float = 1.205, viscosity: float = 1.84e-05, prandtl_number: float = 0.71, heat_capacity_ratio: float = 1.4, atmospheric_pressure: float = 101325.0, ) -> CriticalCouplingResult ``` Solve resonator/slit geometry for perfect absorption at a frequency. Critical coupling (perfect absorption) requires the acoustic surface impedance `Z = T11 / T21` of the rigidly-backed panel to satisfy `Re(Z) cos(theta) = Z0` and `Im(Z) = 0` at `target_frequency` (Appl. Sci. 2017 Eq. (9)), i.e. the reflection zero lies on the real-frequency axis. Holding the neck geometry and cavity side of `resonator` fixed, this tunes the cavity length (which sets the resonance frequency) and the slit height (which sets the visco-thermal leakage balance) to meet both conditions, so `alpha ~ 1` at the design point. **Parameters** | Name | Description | | :--- | :--- | | `target_frequency` | Design frequency `f0`, in hertz. | | `resonator` | Base geometry; its `cavity_length` is used as the initial guess and its neck and cavity side are held fixed. | | `lattice_step` | Resonator lattice step `a`, in metres. | | `period` | Slit array period `d`, in metres. | | `angle` | Design angle of incidence `theta`, in radians. | | `slit_height_bounds` | Search bounds for the slit height, in metres. | | `cavity_length_bounds` | Search bounds for the cavity length, in metres. | | `end_correction` | Include the resonator radiation end corrections. | | `slit_radiation` | Include the slit-to-free-air radiation correction. | | `speed_of_sound` | Speed of sound `c0` in air, in m/s. | | `air_density` | Air density `rho0`, in kg/m3. | | `viscosity` | Dynamic viscosity `eta` of air, in Pa s. | | `prandtl_number` | Prandtl number `Pr` of air. | | `heat_capacity_ratio` | Ratio of specific heats `gamma`. | | `atmospheric_pressure` | Static pressure `P0`, in Pa. | **Returns:** A [`CriticalCouplingResult`](/phonometry/reference/api/materials/slow-sound-absorber/#criticalcouplingresult). A [`SlowSoundAbsorberWarning`](/phonometry/reference/api/materials/slow-sound-absorber/#slowsoundabsorberwarning) is emitted (via `warnings.warn`) if the solver does not reach perfect absorption within tolerance. ## CriticalCouplingResult ```python CriticalCouplingResult( target_frequency: float, angle: float, resonator: HelmholtzResonator, slit_height: float, absorption: float, normalized_impedance: complex, converged: bool, ) ``` Outcome of a critical-coupling (perfect-absorption) design. `resonator` and `slit_height` are the solved geometry that places the reflection zero on the real-frequency axis at `target_frequency` and `angle`; `absorption` is the modelled coefficient there (`~1`) and `normalized_impedance` the achieved `Z cos(theta) / Z0` (`~1`). `converged` flags whether the root find met its tolerance. ## helmholtz_resonator_impedance ```python helmholtz_resonator_impedance( frequency: ArrayLike, resonator: HelmholtzResonator, *, slit_height: float | None = None, lattice_step: float | None = None, end_correction: bool = True, air_density: float = 1.205, viscosity: float = 1.84e-05, prandtl_number: float = 0.71, heat_capacity_ratio: float = 1.4, atmospheric_pressure: float = 101325.0, sum_terms: int = 40, ) -> Complex ``` Acoustic impedance of a Helmholtz resonator with visco-thermal losses. The neck and cavity use the square-duct effective parameters of [`rectangular_duct_properties`](/phonometry/reference/api/materials/slow-sound-absorber/#rectangular_duct_properties); the impedance is Appl. Phys. Lett. 2016 Eq. (A23) with the neck-to-cavity radiation correction of Eq. (A24) and, when `slit_height` and `lattice_step` are supplied, the neck-to-slit correction of Eqs. (A25)-(A26) added to the total neck length correction: `Z_HR = -j [cos(k_n l_n) cos(k_c l_c) - Z_n k_n dl cos(k_n l_n) sin(k_c l_c) / Z_c - Z_n sin(k_n l_n) sin(k_c l_c) / Z_c] / [sin(k_n l_n) cos(k_c l_c) / Z_n - k_n dl sin(k_n l_n) sin(k_c l_c) / Z_c + cos(k_n l_n) sin(k_c l_c) / Z_c]` with `Z_n = sqrt(kappa_n rho_n) / w_n^2`, `k_n = w sqrt(rho_n / kappa_n)` (and likewise for the cavity), reducing to Eq. (A22) when `dl = 0`. **Parameters** | Name | Description | | :--- | :--- | | `frequency` | Frequency vector `f`, in hertz. | | `resonator` | The [`HelmholtzResonator`](/phonometry/reference/api/materials/slow-sound-absorber/#helmholtzresonator) geometry. | | `slit_height` | Slit height `h` for the neck-to-slit correction; if `None` that correction is omitted. | | `lattice_step` | Lattice step `a` for the neck-to-slit correction. | | `end_correction` | Include the radiation end corrections (default True). | | `air_density` | Air density `rho0`, in kg/m3. | | `viscosity` | Dynamic viscosity `eta` of air, in Pa s. | | `prandtl_number` | Prandtl number `Pr` of air. | | `heat_capacity_ratio` | Ratio of specific heats `gamma`. | | `atmospheric_pressure` | Static pressure `P0`, in Pa. | | `sum_terms` | Transverse modes kept per axis in the duct series. | **Returns:** Complex acoustic impedance `Z_HR`, in Pa s/m3, shaped like `frequency`. ## HelmholtzResonator ```python HelmholtzResonator( neck_length: float, neck_side: float, cavity_length: float, cavity_side: float, ) ``` A square-cross-section Helmholtz resonator loading a slit. `neck_length` `l_n` and `neck_side` `w_n` describe the neck, `cavity_length` `l_c` and `cavity_side` `w_c` the closed cavity; all lengths are in metres. ### HelmholtzResonator.plot() ```python HelmholtzResonator.plot( ax: Axes | None = None, *, language: str = 'en', **kwargs: Any, ) -> Axes ``` Draw the resonator cross-section to scale (dimensioned). Requires matplotlib (`pip install phonometry[plot]`); returns the `Axes`. ## plot_helmholtz_resonator_geometry ```python plot_helmholtz_resonator_geometry( resonator: HelmholtzResonator, ax: Axes | None = None, *, language: str = 'en', **kwargs: Any, ) -> Axes ``` Draw a square-section Helmholtz resonator cross-section, to scale. Neck opening upward into free air, cavity below, with the four defining dimensions (neck side and length, cavity side and length) dimensioned. **Parameters** | Name | Description | | :--- | :--- | | `resonator` | A [`HelmholtzResonator`](/phonometry/reference/api/materials/slow-sound-absorber/#helmholtzresonator). | | `ax` | Existing axes, or `None` to create a figure. | | `language` | Label language, `"en"` (default) or `"es"`. | | `kwargs` | Forwarded to the cavity rectangle. | **Returns:** The axes. ## plot_slit_absorber_geometry ```python plot_slit_absorber_geometry( resonators: Sequence[HelmholtzResonator] | HelmholtzResonator, ax: Axes | None = None, *, slit_height: float, lattice_step: float, period: float, language: str = 'en', **kwargs: Any, ) -> Axes ``` Draw one period of the slit metamaterial absorber, to scale. Side cut of the panel: the slit (height `h`) runs from the mouth at the left into the panel; `N` Helmholtz resonators load it from below at the lattice step `a` (total depth `L = N a`); the panel repeats vertically with `period` `d`; rigid back wall at the right. **Parameters** | Name | Description | | :--- | :--- | | `resonators` | The resonator chain of [`slit_helmholtz_absorber`](/phonometry/reference/api/materials/slow-sound-absorber/#slit_helmholtz_absorber) (one per lattice step, or a single resonator reused for all steps). | | `ax` | Existing axes, or `None` to create a figure. | | `slit_height` | Slit height `h`, in metres. | | `lattice_step` | Lattice step `a`, in metres. | | `period` | Panel period `d`, in metres. | | `language` | Label language, `"en"` (default) or `"es"`. | | `kwargs` | Forwarded to the slit rectangle. | **Returns:** The axes. ## rectangular_duct_properties ```python rectangular_duct_properties( frequency: ArrayLike, *, side: float, air_density: float = 1.205, viscosity: float = 1.84e-05, prandtl_number: float = 0.71, heat_capacity_ratio: float = 1.4, atmospheric_pressure: float = 101325.0, sum_terms: int = 40, ) -> tuple[Complex, Complex] ``` Effective density and bulk modulus of a square duct of the given side. The Stinson (1991) rectangular-duct series (Appl. Sci. 2017 Eqs. (7)-(8)), `rho = -rho0 a^2 b^2 / (64 G_rho^2 S_rho)` and `kappa = kappa0 / (gamma + 64 (gamma - 1) G_kappa^2 / (a^2 b^2) S_kappa)`, with `S = sum_k sum_m [alpha_k^2 beta_m^2 (alpha_k^2 + beta_m^2 - G^2)]^-1`, `alpha_k = (2k+1) pi / a`, `beta_m = (2m+1) pi / b`, `G_rho^2 = j w rho0 / eta` and `G_kappa^2 = j w Pr rho0 / eta`. Here the duct is square (`a = b = side`). The series is transcribed in the source's time convention and returned conjugated so the result is passive in the `e^{+j w t}` convention (`Im(k) < 0`). The normalising constant 64 is fixed by the exact limits `rho -> rho0`, `kappa -> kappa0` as the boundary layers vanish and by the Poiseuille resistivity `28.454 eta / side^2` as `w -> 0`. **Parameters** | Name | Description | | :--- | :--- | | `frequency` | Frequency vector `f`, in hertz. | | `side` | Square-duct side length, in metres. | | `air_density` | Air density `rho0`, in kg/m3. | | `viscosity` | Dynamic viscosity `eta` of air, in Pa s. | | `prandtl_number` | Prandtl number `Pr` of air. | | `heat_capacity_ratio` | Ratio of specific heats `gamma`. | | `atmospheric_pressure` | Static pressure `P0`, in Pa. | | `sum_terms` | Transverse modes kept per axis (default 40). | **Returns:** `(rho, kappa)` complex arrays shaped like `frequency`. ## slit_effective_properties ```python slit_effective_properties( frequency: ArrayLike, *, slit_height: float, air_density: float = 1.205, viscosity: float = 1.84e-05, prandtl_number: float = 0.71, heat_capacity_ratio: float = 1.4, atmospheric_pressure: float = 101325.0, ) -> tuple[Complex, Complex] ``` Effective density and bulk modulus of a narrow slit of height `h`. `rho_s = rho0 [1 - tanh(x_rho) / x_rho]^-1` and `kappa_s = kappa0 [1 + (gamma - 1) tanh(x_kappa) / x_kappa]^-1` with `x_rho = (h/2) sqrt(j w rho0 / eta)` and `x_kappa = (h/2) sqrt(j w Pr rho0 / eta)` (Appl. Sci. 2017 Eq. (6); Appl. Phys. Lett. 2016 Eqs. (A1)-(A2)). `kappa0 = gamma P0`. **Parameters** | Name | Description | | :--- | :--- | | `frequency` | Frequency vector `f`, in hertz. | | `slit_height` | Slit height `h`, in metres. | | `air_density` | Air density `rho0`, in kg/m3. | | `viscosity` | Dynamic viscosity `eta` of air, in Pa s. | | `prandtl_number` | Prandtl number `Pr` of air. | | `heat_capacity_ratio` | Ratio of specific heats `gamma`. | | `atmospheric_pressure` | Static pressure `P0`, in Pa. | **Returns:** `(rho_s, kappa_s)` complex arrays shaped like `frequency`. ## slit_helmholtz_absorber ```python slit_helmholtz_absorber( frequency: ArrayLike, resonators: HelmholtzResonator | list[HelmholtzResonator] | tuple[HelmholtzResonator, ...], *, slit_height: float, lattice_step: float, period: float, angle: float = 0.0, end_correction: bool = True, slit_radiation: bool = True, speed_of_sound: float = 343.0, air_density: float = 1.205, viscosity: float = 1.84e-05, prandtl_number: float = 0.71, heat_capacity_ratio: float = 1.4, atmospheric_pressure: float = 101325.0, ) -> SlitResonatorAbsorberResult ``` Transfer-matrix prediction of a slit panel loaded with resonators. The panel is a periodic array (period `d` along the panel face) of thin closed slits of height `h`, each loaded from its upper wall by the given `resonators` spaced by the lattice step `a` (Appl. Sci. 2017, Section 2). The total chain matrix is `T = M_dl (M_s M_HR M_s) ...` over the `N` resonators, where each resonator sits between two half-lattice slit steps; the rigidly-backed reflection factor is `R = (T11 cos(theta) - Z0 T21) / (T11 cos(theta) + Z0 T21)` with `Z0 = rho0 c0 / S0`, `S0 = d a`, and `alpha = 1 - |R|^2` (Eq. (4)). The structure is locally reacting, so the internal chain does not depend on `theta`; only the front air impedance carries `cos(theta)`. **Parameters** | Name | Description | | :--- | :--- | | `frequency` | Frequency vector `f`, in hertz. | | `resonators` | One [`HelmholtzResonator`](/phonometry/reference/api/materials/slow-sound-absorber/#helmholtzresonator) or a sequence of them, ordered from the panel face towards the rigid backing. | | `slit_height` | Slit height `h`, in metres. | | `lattice_step` | Resonator lattice step `a` along the slit, in metres; the slit depth is `L = N a`. | | `period` | Slit array period `d` along the face, in metres (`d >= h`). | | `angle` | Polar angle of incidence `theta`, in radians (`0 <= theta < pi/2 - 1e-6`). | | `end_correction` | Include the resonator radiation end corrections. | | `slit_radiation` | Include the slit-to-free-air radiation correction. | | `speed_of_sound` | Speed of sound `c0` in air, in m/s. | | `air_density` | Air density `rho0`, in kg/m3. | | `viscosity` | Dynamic viscosity `eta` of air, in Pa s. | | `prandtl_number` | Prandtl number `Pr` of air. | | `heat_capacity_ratio` | Ratio of specific heats `gamma`. | | `atmospheric_pressure` | Static pressure `P0`, in Pa. | **Returns:** A [`SlitResonatorAbsorberResult`](/phonometry/reference/api/materials/slow-sound-absorber/#slitresonatorabsorberresult). ## SlitResonatorAbsorberResult ```python SlitResonatorAbsorberResult( frequency: Real, angle: float, surface_impedance: Complex, normalized_impedance: Complex, reflection: Complex, absorption: Real, effective_wavenumber: Complex, effective_impedance: Complex, transfer_matrix: Complex, resonators: tuple[HelmholtzResonator, ...] | None = None, slit_height: float | None = None, lattice_step: float | None = None, period: float | None = None, ) ``` Prediction of a slit panel loaded with Helmholtz resonators. All spectra share the shape of `frequency`. `surface_impedance` is the acoustic surface impedance `Z = T11 / T21` in Pa s/m3 of the rigidly backed panel, `normalized_impedance` its ratio to `Z0 = rho0 c0 / S0`, `reflection` the plane-wave reflection factor `R(theta)`, `absorption` the coefficient `alpha = 1 - |R|^2`, `effective_wavenumber` and `effective_impedance` the retrieved `k_eff` and `Z_eff` (Appl. Sci. 2017 Eq. (5)), and `transfer_matrix` the total 2x2 chain matrix with shape `(2, 2, len(frequency))`. The trailing fields retain the panel geometry the prediction was run with (`resonators`, `slit_height`, `lattice_step`, `period`) so `plot_geometry` can draw the cross-section; they are appended after the original fields and default to `None` for hand-built results. ### SlitResonatorAbsorberResult.plot() ```python SlitResonatorAbsorberResult.plot( ax: Axes | None = None, *, language: str = 'en', **kwargs: Any, ) -> Axes ``` Plot the absorption spectrum `alpha(f)` with `|R|` overlaid. Requires matplotlib (`pip install phonometry[plot]`); returns the `Axes`. ### SlitResonatorAbsorberResult.plot_geometry() ```python SlitResonatorAbsorberResult.plot_geometry( ax: Axes | None = None, *, language: str = 'en', **kwargs: Any, ) -> Axes ``` Draw one period of the panel cross-section to scale (dimensioned). Requires matplotlib (`pip install phonometry[plot]`); returns the `Axes`. **Raises** | Exception | When | | :--- | :--- | | ValueError | If the result does not retain its geometry. | ## SlowSoundAbsorberWarning Advisory for slow-sound absorber use outside the modelled regime.